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Mirrors > Home > MPE Home > Th. List > lsppreli | Structured version Visualization version GIF version |
Description: A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lsppreli.v | ⊢ 𝑉 = (Base‘𝑊) |
lsppreli.p | ⊢ + = (+g‘𝑊) |
lsppreli.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lsppreli.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lsppreli.k | ⊢ 𝐾 = (Base‘𝐹) |
lsppreli.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsppreli.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsppreli.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lsppreli.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
lsppreli.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsppreli.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lsppreli | ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ (𝑁‘{𝑋, 𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsppreli.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lsppreli.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | lsppreli.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lsppreli.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 3, 4 | lspsnsubg 19735 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
6 | 1, 2, 5 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
7 | lsppreli.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
8 | 3, 4 | lspsnsubg 19735 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
9 | 1, 7, 8 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
10 | lsppreli.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | lsppreli.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
12 | lsppreli.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
13 | lsppreli.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
14 | 3, 10, 11, 12, 4, 1, 13, 2 | lspsneli 19756 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
15 | lsppreli.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
16 | 3, 10, 11, 12, 4, 1, 15, 7 | lspsneli 19756 | . . 3 ⊢ (𝜑 → (𝐵 · 𝑌) ∈ (𝑁‘{𝑌})) |
17 | lsppreli.p | . . . 4 ⊢ + = (+g‘𝑊) | |
18 | eqid 2821 | . . . 4 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
19 | 17, 18 | lsmelvali 18758 | . . 3 ⊢ ((((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) ∧ ((𝐴 · 𝑋) ∈ (𝑁‘{𝑋}) ∧ (𝐵 · 𝑌) ∈ (𝑁‘{𝑌}))) → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
20 | 6, 9, 14, 16, 19 | syl22anc 836 | . 2 ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
21 | 3, 4, 18, 1, 2, 7 | lsmpr 19844 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
22 | 20, 21 | eleqtrrd 2916 | 1 ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ (𝑁‘{𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4553 {cpr 4555 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 +gcplusg 16548 Scalarcsca 16551 ·𝑠 cvsca 16552 SubGrpcsubg 18256 LSSumclsm 18742 LModclmod 19617 LSpanclspn 19726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-cntz 18430 df-lsm 18744 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-lmod 19619 df-lss 19687 df-lsp 19727 |
This theorem is referenced by: lspexch 19884 baerlem3lem1 38875 baerlem5alem1 38876 baerlem5blem1 38877 |
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